Symmetry-Based Indicators of Band Topology I: Formalism*

We develop an efficient theory to answer one of the oldest questions in band theory, namely, what are the possible symmetry-respecting ways to connect energy bands across the Brillouin zone? We find solutions for all 1,651 magnetic space groups in 3D, from which we extract fundamental physical quantities, like the minimal number of electrons required to form a band insulator, and how nontrivial band topology may be reflected in the symmetry eigenvalues.